The** history of geometry** starts with the first civilizations to make use of this branch of mathematics in a practical way, specifically the peoples of the Indus Valley and Babylonia who knew of obtuse triangles, around 3000 BC

Methods for calculating the area of a circle are used in the writings of the Egyptian scribe Ahmes (1550 BC). For their part, the Babylonians had general rules for measuring volumes and areas.

Versions of the Pythagorean theorem were known to both the Egyptians and Babylonians 1,500 years before the Pythagorean versions. On the other hand, the Indians of the Vedic period (1500-100 BC) used geometry in the construction of altars.

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**Ancient Greece**

The Greeks were inclined towards the development of mathematics for a long time. Characters like Pythagoras and Plato, related numbers to everything that exists in the world. For them mathematics was the key to interpret the universe; this ideal continued in the followers of Pythagoreans for several centuries.

**Thales of Miletus**

Thales of Miletus was one of the first Greeks to contribute to the advances in geometry. He spent a lot of time in Egypt and from this he learned the basics. He was the first to establish formulas for measuring geometry.

He managed to measure the height of the Egyptian pyramids, measuring its shadow at the exact moment when its height was equal to the measure of its shadow.

**Pythagoras**

Within the most significant contribution of Pythagoras (569 BC – 475 BC) to geometry is the famous Pythagorean theorem, which states that within a right triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the remaining sides.

**Euclid’s Elements**

The most outstanding work that has been able to be rescued from antiquity has been the study The elementsby Euclid of Alexandria (325 BC – 265 BC), made during the years 300 BC It is a work of great historical value that has served as the basis for the teaching of mathematics for more than 2000 years.

*The elements* it was one of the first books and compilations of studies that explained mathematical principles that could be applied to any situation. It includes the postulates, which are the fundamental principles of geometry in his work. On the other hand, there are the quantitative principles known as the basic notions.

Euclid’s teaching reduced the construction instruments within geometry to only two: a ruler without measures and a compass. This generated the three classic problems that found no answers until the 19th century: the squaring of the circle, the doubling of the cube, and the trisection of an angle.

For the ancients, the two ideal geometric entities were the straight line and the circle, so most of the geometric theorems that were proposed were the product of exploration with these instruments.

**geometry in astronomy**

Geometry was also useful for the Greeks in terms of the study of the stars. They calculated the movements through observation and made geometric plans of the sky establishing the Earth as a central point, and both the Sun and the Moon and the other planets as entities that moved around it, rotating in a set of circles.

One of the most influential contributions was the *Almagest*, written in the 2nd century AD by Claudius Ptolemy (AD 100-AD 170), an astronomical treatise containing the catalog of the stars. It was the most complete text of his time and had a transcendental influence on astronomical studies until very late in the Middle Ages. He was part of the media that most popularized the geocentric system, which affirmed that the Earth was the center of the universe.

**Islamic influence**

By the 9th century, when the Arab world was in its great expansion, much of their culture permeated various areas of science and the arts. They were great admirers of the mathematical and philosophical works of the Greeks.

One of the most explored branches within their needs was astronomy, in order to locate the exact orientation in which Mecca was located in order to perform prayers.

Following the studies of Euclid and other contributions such as those of Ptolemy, the Muslims developed the stereographic projection, that is, the projection of the celestial sphere on the plane to be used as a map. This meant progress in terms of the study of trigonometry.

Among the most representative characters is Thābit ibn Qurra (826/36-901) who made relevant translations of the ancient texts of Apollonius, Archimedes, Euclid and Ptolemy. Some of these are the only surviving versions of the ancient scriptures.

The explorations in terms of astronomical geometry also allowed the creation of one of the most representative instruments, the astrolabe, which simplified the astronomical calculations of the moment. In addition, this instrument also allowed them to know the time and finally get the orientation to Mecca.

**Development of the European legacy**

In the twelfth century, after the insertion of the classical teachings of the Greeks thanks to the Muslim expansion and the development of their own discoveries, translations of the texts into Latin began directly from Greek or from the Arabic language.

This would open the way for a new learning within Europe that would be fueled by the Renaissance. The rediscovery of notions such as «proofs» began, a concept developed among the Greeks who were interested in the demonstration of postulates in reality.

**Geometry in art**

Knowledge was also reflected in the arts, such as painting or architecture, since geometry would begin to be a fundamental part of the development of perspective in art.

Filippo Brunelleschi (1377-1446), was the one who managed to develop linear perspective through mathematics. The objective of this theory was to be able to represent a three-dimensional space on a plane based on how it was perceived by the human eye. Thus, he establishes that all the lines in a painting had to converge or meet at a vanishing point in order to generate the sensation of depth.

Brunelleschi was the first to describe perspective as a scientific procedure, and this served as the foundation for later work within the arts.

Among other examples of the application of geometry to the study of art and of the human being itself, is the work of Leonardo da Vinci (1452-1519) in his drawing The Vitruvian Man. It is a study based on the most perfect proportions for the human body through a geometric analysis of its structure.

Among other areas, architecture also stands out, where various elements such as symmetry and balance began to appear as fundamental characteristics. Square, rectangular doors and windows, positioned in a balanced way; use of classical elements of antiquity such as columns, domes and vaults.

**step to modernity**

The analysis of perspectives and projections during the Renaissance was one of the incentives to arouse the interest of mathematicians. From this moment on, more solid and complex mathematical bases begin to be founded within geometry.

One of the most important works for modernity was that of the architect Girard Desargues (1591–1661), which marked the beginning of projective geometry. He, on the one hand, established that the parallel lines in a projection should converge at a point on the line of infinity, that is, the horizon.

On the other hand, he also discovered what would be recognized as Desargues’ theorem, which establishes the relationship between two figures that can be considered «projective». He was also in charge of simplifying the works of Apolonio with respect to the sections of a cone, making analogies between this figure and the cylinder.

Another great event of the period was the creation of analytical geometry through the studies of René Descartes (1596-1650) and Pierre de Fermat (1601-1665) independently. It is the study of geometry through the use of a coordinate system.

**non-euclidean geometry**

Towards the 18th and 19th centuries, studies began that led to non-Euclidean geometry. Specifically, it was Gauss, Johann Bolyai, and Lobachevsky who verified that Euclid’s fifth postulate, known as the parallel postulate, could not be verified.

In this way they developed a type of geometry in which this postulate was classified as false. This new form was successful in giving satisfactory results in styles of geometry that did not necessarily comply with all of Euclid’s postulates. This is how hyperbolic geometry and elliptical geometry were later born.

It is worth highlighting the work of Leonhard Euler (1707-1783) in the 18th century, regarding the development of mathematical notation. Subsequently, the 20th century would bring with it the development of more specific fields of geometry, among which are:

–Algebraic geometry: It is a branch of mathematics that combines abstract algebra and analytical geometry.

–The finite geometry: It is a geometric system that is made up of a finite number of points, that is, they have an end or limit and therefore can be measured.

–digital geometry: It is a branch of computer science that carries out the study of algorithms and data structures that can be represented in geometric terms.

**References**

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History of geometry. Wikipedia, the free encyclopedia. Retrieved from en.wikipedia.org

Analytic geometry. Wikipedia, the free encyclopedia. Retrieved from en.wikipedia.org

(2017) Geometry And Mathematics In The Renaissance. Recovered from ukessays.com

Sáiz A. Linear Perspective in Brunelleschi. University of Valencia. Recovered from uv.es

The Editors of Encyclopaedia Britannica (2019). Renaissance architecture. Encyclopædia Britannica. Recovered from britannica.com

AndersenK (2020). Girard Desargues. Encyclopædia Britannica. Recovered from britannica.com

(2011) An Interesting Introduction to computational geometry. Recovered from gaussianos.com